STUPIDSID
Solve any two question Q.1(a)(i, ii, iii) Solve any one question from Q.1(a,b) & Q.2(a,b,c)
1(a)(i)
\[\left ( D^{2}-7D+6 \right )y=e^{2x}\]
4 M
1(a)(ii)
\[\left ( D^{2} +4\right )y = x\sin x\]
4 M
1(a)(iii)
\[\left ( x+2 \right )^{2}\frac{d^{2}y}{dx^{2}}+3\left ( x+2 \right )\frac{dy}{dx}+y=4\sin \log \left ( x+2 \right )\]
4 M
1(b)
Find the Fourier sine transform of the function: \[f(x)=e^{-x},x> 0.\]
4 M
2(a)
A circuit consists of an inductance L and condenser of capacity C in series. An alternating e.m.f. E sin n t is applied to it at time t = 0, the initial current and charge on the condenser being zero and \(w^{2}=\frac{1}{LC}, \)/ find the current flowing in the circuit at any time for w≠n.
4 M
Solve any two question Q.2(b)(i, ii)
2(b)(i)
Find inverse z-transform \[F(z)=\frac{z}{\left ( z-1 \right )\left ( z-3 \right )},|z|> 3\]
4 M
2(b)(ii)
\( F(z)=\frac{z}{\left ( z-\frac{1}{4} \right )\left ( z-\frac{1}{5}\right )}.\)/ (by inversion integral method).
4 M
2(c)
Solev the following difference equation:\( \begin{align*}f\left ( k+1 \right )+\frac{1}{4}f(k)=\left ( \frac{1}{4} \right )^{k};\\
f(0)=0, k\geq 0 \end{align*}
\)/
4 M
Solve any one question from Q.3(a,b,c) &Q.4(a,b,c)
3(a)
Using fourth order Runge-Kutta method, solve the differential equation: \( \frac{dy}{dx}=x+y+xy \)/ with y(0) = 1 to get y(0, 1) taking h = 0.1
4 M
3(b)
Find Lagrange's interpolating polynomial passing through the set of points:
| x | y |
| 0 | 3 |
| 1 | 4 |
| 3 | 12 |
4 M
3(c)
Find the directional derivative of: \( \phi =x^{2}-y^{2}+2z^{2} \)/ at the point (1, 2, 3) in the direction of \[4\bar{i}-2\bar{j}+\bar{k}.\]
4 M
Solve any two question Q.4(a)(i, ii)
4(a)(i)
Show that: \[\nabla\left ( \frac{\bar{a}.\bar{r}}{r^{2}} \right )=\frac{\bar{a}}{r^{2}}-\frac{2\left ( \bar{a}.\bar{r} \right )\bar{r}}{r^{4}}\]
4 M
4(a)(ii)
\[\nabla\left ( \bar{a} .\nabla\frac{1}{r}\right )=\frac{3\left ( \bar{a}.\bar{r} \right )\bar{r}}{r^{5}}-\frac{\bar{a}}{r^{3}}\]
4 M
4(b)
Show that : \( \bar{F}=\left ( 6xy+z^{3} \right )\bar{i}+\left ( 3x^{2}-z \right )\bar{j}+\left ( 3xz^{2} -y\right )\bar{k}\)/ is irrotational. Find the scalar pothential ϕ such that \[\bar{F}=\nabla\phi .\]
4 M
4(c)
By using Trapezoidal rule, evaluate:\(\int_{0}^{2}\frac{1}{1+x^{4}}\ dx \)/ taking \(h=\frac{1}{2}, \)/ correct upto 3-decimal places.
4 M
Solve any two question from Q.5(a,b,c) & Solve any one question from Q.5(a, b,c) &Q.6(a, b, c)
5(a)
Evaluate \(\int \bar{F}.d\bar{r}\ \text{bar}\\
\
\bar{F}=3x^{2}\ \bar{i}+\left ( 2xy-y \right )\bar{j}+z\bar{k} \)/ also straight line joining (0, 0, 0) and (2, 1, 3).
4 M
5(b)
Evaluate : \( \iint _{S}\left [ \left ( 4x+3yz^{2} \right ) \bar{i}-\left ( x^{2}z^{2} +y\right )\bar{j}+\left ( y^{2} +2z\right )\bar{k}\right ].d\bar{S} \)/ where S is the surface of the sphere \[x^{2}+y^{2}+z^{2}=9.\]
4 M
5(c)
Apply Stokes's theorem to evaluate \( \oint _{C}\bar{F}.d\bar{r} \)/ where: \( \bar{F}=xy^{2}\bar{i}+y\bar{j}+xz^{2}\bar{k} \)/ and C is the boundary of reactangle:
x=0, y=0, x=1, y=2 in z=0 plane.
x=0, y=0, x=1, y=2 in z=0 plane.
5 M
6(a)
Using Green's lemma evaluate: \( \int _{C}\left ( xy-x^{2} \right )dx+x^{2}y \ \ dy\)/ along the curve C : x=1, y=x, y=0.
4 M
6(b)
Evaluate: \( \iint \left ( \nabla\times \bar{F} \right ).\hat{n} \ \ ds, \)/ where \( \bar{F}=\left ( x-y \right )\bar{i}-\left ( x^{2}+yz \right )\bar{j}-3xy^{^{2}}\bar{k} \)/ and S is the surface of the cone \( z=4-\sqrt{x^{2}+y^{2}},\)/ above the xoy plane.
5 M
6(c)
Prove that:\[\iint _{s}\left ( \phi \nabla\Psi -\Psi \nabla\phi \right ).d\bar{S}=\iint _{v}\int \left ( \phi \nabla^{2} \Psi -\Psi \nabla^{2}\phi \right )dV.\]
4 M
Solve any one question from Q.7(a,b,c) & Q.8(a,b,c)
7(a)
Show that the function: \( f(z)= u+iv\)/ with constant modulus and constant amplitude is constant in each case.
4 M
7(b)
Evaluate:\( \oint _{C}\frac{4z^2+z}{z^{2}-1}\ \ dz \)/ where C is the circle |z-1| = \[\frac{1}{2}\].
4 M
7(c)
Find the bilinear tranformation which maps the points: z=1, i, 2i onto the points w = -2i, 0, 1 respecitvely.
5 M
8(a)
If f(z) = u+iv is analytic, find f(z), if\(u-v= x^{2}-y^{2}-2xy. \)/
5 M
8(b)
Evaluate: \( \oint _C\frac{\left ( z^{2} +cos^{2}z\right )}{\left ( z-\frac{\pi }{4} \right )}^{3} dz, \)/ where C is circle |z| =1
4 M
8(c)
Find the map of the straight line y=x under the transformation \[w=\frac{z-1}{z+1}.\]
4 M
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